integral of (110)/((x+1)(x^2+9)^2)

解答

\int \frac{110}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

110 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}) + C

求解步骤

\int \frac{110}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 110 \cdot \int \frac{1}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

将

\frac{1}{( x + 1 ) ( x ^{2} + 9 ) ^{2}}

用部份分式展开:

\frac{1}{100 ( x + 1 )} + \frac{- x + 1}{100 ( x ^{2} + 9 )} + \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}}

= 110 \cdot \int \frac{1}{100 ( x + 1 )} + \frac{- x + 1}{100 ( x ^{2} + 9 )} + \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 110 (\int \frac{1}{100 ( x + 1 )} d x + \int \frac{- x + 1}{100 ( x ^{2} + 9 )} d x + \int \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}} d x)

\int \frac{1}{100 ( x + 1 )} d x = \frac{1}{100} ln ∣ x + 1 ∣

\int \frac{- x + 1}{100 ( x ^{2} + 9 )} d x = \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}))

\int \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}} d x = \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}

= 110 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )})

解答补常数

= 110 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}) + C

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